Local Fractional Derivative Research Articles

A new technique of reduce differential transform method to solve local fractional PDEs in mathematical physics

In this manuscript, we investigate solutions of the partial differential equations (PDEs) arising in mathematical physics with local fractional derivative operators (LFDOs). To get approximate solutions of these equations, we utilize the reduce differential transform method (RDTM) which is based upon the LFDOs. Illustrative examples are given to show the accuracy and reliable results. The obtained solutions show that the present method is an efficient and simple tool for solving the linear and nonlinear PDEs within the LFDOs.

Thermography of ferromagnetic Walter's-B fluid through varying thermal stratification

Thermally induced damages (hypolimnion, metalimnion and epilimnion) are usually caused by thermal stratification; due to a fact that fluid flows at a low velocity with large temperature variances. In this manuscript, thermal stratification phenomenon is studied for unsteady flow of Walter's-B fluid subject to a consistent inclined magnetic field at an angle of inclination over the boundary of transverse xy-plane with fractional thermal transport. Non-singular and local fractional derivative is hauled in the constitutive equations for the thermal flux. Semi analytical solutions of the dimensionless temperature and velocity fields are recognized by using the Laplace inversion numerical algorithms Stehfest's and Tzou's. Temperature and rate of heat transfer for non-integer and integer order derivatives are computed. Finally, the influence of some flow parameters and fractional parameter α on the temperature and velocity fields are graphically underlined by using IMATHCAD software and discussed.

On the nondifferentiable exact solutions to Schamel's equation with local fractional derivative on Cantor sets

AbstractThe various aspects of differential calculus are always on the path to progress and excellence, and these trends have been more highlighted in recent decades. More specifically, tremendous advances have been made in the field of fractional calculus. One of the main branches in this field is the local fractional derivative, which has been used successfully to describe many real‐world phenomena in science, and engineering. The present contribution aims to adopt a newly proposed analytical technique to construct exact solutions to local fractional Schamel's equation defined on Cantor sets. To this end, a set of elementary functions are constructed on Cantor sets is defined. Furthermore, the combination of these modified functions is utilized to constitute a formal representation of the searched exact solution for the equation. Numerical simulations related to some of the obtained solutions are also included. The acquired results confirm that the method used is not only very straightforward but also efficient in terms of application. In order to perform complicated and tedious calculations while solving this problem, the use of a symbolic computational packages in Maple or Mathematica is inevitable. The work emphasizes the power of the employed method in providing various exact solutions to different physical problems involving local fractional derivatives.

Equation Including Local Fractional Derivative and Neumann Boundary Conditions

The aim of this study to discuss the construction of the solution of fractional partial differential equations (FPDEs) with initial and boundary conditions. Since the homogenous initial boundary value problem involves local fractional-order derivative, it has classical initial and boundary conditions. By means of the separation of variables method (SVM) and the inner product on L^2\left[0,l\right], we construct the solution in this series form in terms of eigenfunctions of related Sturm-Liouville problem. An illustrative example presents the applicability and influence of the separation of variables method on fractional mathematical problems.

On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique

One of the most interesting branches of fractional calculus is the local fractional calculus, which has been used successfully to describe many fractal problems in science and engineering. The main purpose of this contribution is to construct a novel efficient technique to retrieve exact fractional solutions to local fractional Gardner's equation defined on Cantor sets by an effective numerical methodology. In the framework of this technique, first a set of elementary functions are defined on the contour set. Taking these functions into account, the general structure of the exact solution for the equation is suggested as a specific combination of the defined basis functions. By determining the unknown coefficients in this expansion and by placing them in the original equation, specific forms of solutions to the fractional equation are determined. Several interesting numerical simulations of the achieved results are also presented in the article to give a better understanding of the dynamic behavior of these results. The results obtained in this research confirm that the method used is very simple and efficient in terms of application. Moreover, it is accurate and free of any errors in terms of calculation. Therefore, it can be employed to handle other partial differential equations including local fractional derivatives.

Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property

In the paper, we extend some previous results dealing with the Hermite–Hadamard inequalities with fractal sets and several auxiliary results that vary with local fractional derivatives introduced in the recent literature. We provide new generalizations for the third-order differentiability by employing the local fractional technique for functions whose local fractional derivatives in the absolute values are generalized convex and obtain several bounds and new results applicable to convex functions by using the generalized Hölder and power-mean inequalities.As an application, numerous novel cases can be obtained from our outcomes. To ensure the feasibility of the proposed method, we present two examples to verify the method. It should be pointed out that the investigation of our findings in fractal analysis and inequality theory is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

An efficient computational approach for local fractional Poisson equation in fractal media

AbstractIn this article, we analyze local fractional Poisson equation (LFPE) by employing q‐homotopy analysis transform method (q‐HATM). The PE describes the potential field due to a given charge with the potential field known, one can then calculate gravitational or electrostatic field in fractal domain. It is an elliptic partial differential equations (PDE) that regularly appear in the modeling of the electromagnetic mechanism. In this work, PE is studied in the local fractional operator sense. To handle the LFPE some illustrative example is discussed. The required results are presented to demonstrate the simple and well‐organized nature of q‐HATM to handle PDE having fractional derivative in local fractional operator sense. The results derived by the discussed technique reveal that the suggested scheme is easy to employ and computationally very accurate. The graphical representation of solution of LFPE yields interesting and better physical consequences of Poisson equation with local fractional derivative.

Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications

The present article addresses the concept of p-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers are p-convex. The method we present is an alternative in showing the classical variants associated with generalized p-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics.

A [formula omitted]-order R-L high-pass filter modeled by local fractional derivative

As an important electronic device, filter is applied to all kinds of electronic products. In this paper, a new ℘-order R-L High-pass filter (HPF) modeled by the local fractional derivative (LFD) is proposed for the first time. With the help of the local fractional Laplace transform (LFLT), we obtain the non-differentiable(ND) transfer function, and present the expressions of ND amplitude-frequency characteristic (AFC) and ND phase-frequency characteristics (PFC). The corresponding parameters and properties of the ℘-order R-L HPF are also studied. What’s interesting is that the ℘-order R-L HPF becomesthe ordinary one in the exceptional case at ℘ = 1. The obtained results in this paper reveal the sufficiency of the local fractional derivative for analyzing the circuit systems in fractal space.

The transient analysis for zero-input response of fractal RC circuit based on local fractional derivative

Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems.

A NEW PERSPECTIVE TO STUDY THE THIRD-ORDER MODIFIED KDV EQUATION ON FRACTAL SET

In this paper, we construct the Bäcklund transformations and the super-position formulas to the constant coefficients local fractional Riccati equation for the first time. Next, by means of the Bäcklund transformations and seed solutions which have been known in [X. J. Yang et al., Non-differentiable solutions for local fractional nonlinear Riccati differential equations, Fundam. Inform. 151(1–4) (2017) 409–417], we can get a class of exact solutions to the third-order modified KdV equation on the fractal set. These new type solutions can assist us to review different nonlinear phenomena better, which had been modeled via local fractional derivative.

Local fractional Moisil–Teodorescu operator in quaternionic setting involving Cantor‐type coordinate systems

The Moisil‐Teodorescu operator is considered to be a good analogue of the usual Cauchy–Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in . In the present work, a general quaternionic structure is developed for the local fractional Moisil–Teodorescu operator in Cantor‐type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil–Teodorescu operator.

A new analysis for Klein-Gordon model with local fractional derivative

This work adopts Yang’s local fractional derivative to define the fractional Klein-Gordon equation in a fractal space or microgravity space. The variational principle of local fractional Klein-Gordon equation is successfully established via fractional semi-inverse transform method and the classical He’s variational iteration method (HVIM) is used to obtain its approximate analytical solution.

On the (N+1)‐dimensional local fractional reduced differential transform method and its applications

In this paper, we generalize the (N+1)‐dimensional local fractional reduced differential transform method (LFRDTM) within the local fractional derivative sense. First, we show some new properties, lemmas, theorems and corollariesfor the (N+1)‐dimensional LFRDTM. Second, these new properties, lemmas and theorems can be proved immediately after. Thirdly, we used two examples to state that this approach is efficient and simple to find numerical solutions to higher dimensional local fractional partial differential equations. Finally, we can be seen that this work can be looked as an extension of the prior work.

NEWTON’S-TYPE INTEGRAL INEQUALITIES VIA LOCAL FRACTIONAL INTEGRALS

We firstly establish an identity involving local fractional integrals. Then, with the help of this equality, some new Newton-type inequalities for functions whose the local fractional derivatives in modulus and their some powers are generalized convex are obtained. Some applications of these inequalities for Simpson’s quadrature rules and generalized special means are also given.

Local fractional Newton\u2019s inequalities involving generalized harmonic convex functions

A new auxiliary result based on a three step quadratic kernel utilizing the concepts of local fractional calculus is obtained. Using this new auxiliary result we have several new Newton type inequalities whose power q of local fractional derivative in modulus is a generalized harmonic convex function.

ON A HIGH-PASS FILTER DESCRIBED BY LOCAL FRACTIONAL DERIVATIVE

The local fractional derivative (LFD) has gained much interest recently in the field of electrical circuits. This paper proposes a non-differentiable (ND) model of high-pass filter described by the LFD, where the ND transfer function is obtained with the help of the local fractional Laplace transform, and its parameters and properties are studied. The obtained results reveal the sufficiency of the LFD for analyzing circuit systems in fractal space.

Fractional derivative for interpolation in [formula omitted] and SO(n) applications in functionally graded materials and rigid body transformations

An integration of fractional calculus, data interpolation, and Lie group leads to new treatments of interpolation problems. A local and centered fractional derivative is introduced, which enables the fractional interpolant to create a Cα-smooth (α∈R+) curve by extending integer order cost functions to fractional ones. The fractional interpolant fills the gaps between the traditional linear interpolating segments, cubic splines, and higher-order splines. Interpolating rigid body transformations has enormous applications in mechanics, computer graphics, and robotics. Previous approaches have provided the re-parametrization methods, the recursive methods (e.g., de Casteljau’s algorithm), and the gradient methods. To approximate the correct interpolating curve on SO(3), a closed-form model that merges the tangent spaces of knots is proposed. The model first constructs a family of candidate curves and then combines them into an improved approximation. Applications in modeling functionally graded materials and rigid body transformations are discussed.

A new analytical solution of Klein–Gordon equation with local fractional derivative

The work presented in this paper is to combine the Sumudu transform method with a variational iteration method for solving linear and nonlinear partial differential equations with local fractional derivative. We apply the proposed method to obtain approximate analytical solutions of Klein–Gordon equations with local fractional derivative. The results obtained for the local fractional Klein–Gordon equation in the three cases presented, are in the form of non-differentiable functions. Through this work it can be said that this method is an alternative analytical method for linear and nonlinear local fractional partial differential equations.

A new class of travelling wave solutions for local fractional diffusion differential equations

In this paper, we investigate a 3-D diffusion equation within the scope of the local fractional derivative. For this model, we establish local fractional vector operators and a local fractional Laplace operator defined on Cantor-type cylindrical coordinate and Cantor-type spherical coordinate, respectively. With the help of the spherical symmetry method based on those operators, we obtain exact traveling wave solutions of the 3-D diffusion equation. The results reveal that the proposed schemes are very effective for obtaining nondifferentiable solutions of fractional diffusion problems.